Extrusion along a curve for a certain face
Various Options .. You may only need one


Use the vertex group entry of the Curve Modifier to Limit the area  of shape influence. 
Use proportional edit
Use an armature to control the shape of an
extrusion.
Consider Spline IK Constraint




See tutorials on any one of these suggestions. Blender Stack Exchange BSE, can be searched on these topics. You can also search the famous video website that has a name that rhymes with noonoob. This was to list various options and is not a tutorial on any of the ideas.